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In [[mathematics]], a [[linear map]] is a [[functionFunction (mathematics)|mapping]] {{<math|''>X''&nbsp;→&nbsp;'' \to Y''}}</math> between two [[Module (mathematics)|module]]s (including [[vector space]]s) that preserves the operations of addition and [[scalarScalar (mathematics)|scalar]] multiplication.
 
By studying the linear maps between two modules one can gain insight into their structures. If the modules have additional structure, like [[topologyTopology|topologies]] or [[Bornological space|bornologies]], then one can study the subspace of linear maps that preserve this structure.
 
== Topologies of uniform convergence on arbitrary spaces of maps ==
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Throughout we assume the following:
<ol>
<li><math>T</math> is any non-empty set and <math>\mathcal{G}</math> is a non-empty collection of subsets of <math>T</math> [[Directed set|directed]] by subset inclusion (i.e. for any <math>G, H \in \mathcal{G}</math> there exists some <math>K \in \mathcal{G}</math> such that {{<math|''>G'' ∪\cup ''H'' ⊆\subseteq ''K''}}</math>).</li>
<li><math>Y</math> is a [[topological vector space]] (not necessarily Hausdorff or locally convex) and {.</li>
<li><math>\mathcal{N}</math|𝒩}}> is a basis of neighborhoods of 0 in <math>Y.</math></li>
<li><math>Y^TF</math> denotesis thea setvector subspace of all <math>Y</math>-valued^T functions= with\prod_{t ___domain\in <math>T.} Y,</math></liref group=note>Because <li><math>FT</math> is just a set that is not yet assumed to be endowed with any vector subspacespace ofstructure, <math>F \subseteq Y^T</math> (should not necessarilyyet consistingbe assumed to consist of linear maps), which is a notation that currently can not be defined.</ref> denotes the set of all <math>Y</math>-valued functions <math>f : T \to Y</math> with ___domain <math>T.</math></li>
</ol>
 
:'''Definition and notation''': For any subsets <math>G</math> of\subseteq <math>X</math> and <math>N</math> of\subseteq <math>Y,</math> let
:"{{<math|1 display=𝒰"block">\mathcal{U}(''G'', ''N'') := \{ ''f'' ∈\in ''F'' : ''f'' (''G'') ⊆\subseteq ''N''}} \}.</math>
 
=== Basic neighborhoods at the origin ===
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;Properties
 
<math>\mathcal{U}(G, N)</math> is an [[Absorbing set|absorbing]] subset of <math>F</math> if and only if for all <math>f \in F,</math> <math>N</math> absorbs <math>f(G)</math>.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
<ul>
<li>If <math>\mathcal{U}(G, N)</math> is an [[AbsorbingBalanced set|absorbingbalanced]] subset of <math>F</math> if and only if for all <math>f \in F,</math> <math>N</math> absorbs {{math|''f'' (''G'')}}.{{sfn|Narici|Beckenstein|2011|pp=371-423}} (respectively, [[Convex set|convex]]) then so is <math>\mathcal{U}(G, N).</limath>
<li>If <math>N</math> is [[Balanced set|balanced]] then so is <math>\mathcal{U}(G, N).</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>If <math>N</math> is [[Convex set|convex]] then so is <math>\mathcal{U}(G, N).</math></li>
</ul>
 
;Algebraic relations
 
<ul>
<li>ForIf any<math>s</math> is a scalar {{mvar|s}},then {{<math|1=''>s''𝒰 \mathcal{U}(''G'', ''N'') = 𝒰\mathcal{U}(''G'', ''sN''s N)}};,</math> so that in particular, {{<math|1=>-𝒰 \mathcal{U}(''G'', ''N'') = 𝒰\mathcal{U}(''G'', -'' N'')}}.</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>{{<math|𝒰>\mathcal{U}(''G'' ∪\cup ''H'', ''M'' ∩\cap ''N'') ⊆\subseteq 𝒰\mathcal{U}(''G'', ''M'') ∩\cap 𝒰\mathcal{U}(''H'', ''N'')}}</math> for any subsets <math>G</math>, and <math>H</math> of\subseteq <math>X</math> and any non-empty subsets {{mvar|M}} and <math>M, N</math> of\subseteq <math>Y.</math>{{sfn|Jarchow|1981|pp=43-55}} Thus:For such subsets, it follows that,
<ul>
<li>Ifif {{<math|''>M'' ⊆\subseteq ''N''}}</math> then <math>\mathcal{{math|𝒰U}(''G'', ''M'') ⊆\subseteq 𝒰\mathcal{U}(''G'', ''N'')}}.</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>Ifif {{<math|''>G'' ⊆\subseteq ''H''}}</math> then <math>\mathcal{{math|𝒰U}(''H'', ''N'') ⊆\subseteq 𝒰\mathcal{U}(''G'', ''N'')}}.</math></li>
<li>For any {{<math|''>M'', ''N'' ∈\in 𝒩}\mathcal{N}</math> and subsets {{<math|''>G'', ''H'', ''K''}}</math> of <math>T,</math> if {{<math|''>G'' ∪\cup ''H'' ⊆\subseteq ''K''}}</math> then {{<math|𝒰>\mathcal{U}(''K'', ''M'' ∩\cap ''N'') ⊆\subseteq 𝒰\mathcal{U}(''G'', ''M'') ∩\cap 𝒰\mathcal{U}(''H'', ''N'')}}.</math></li>
</ul>
</li>
<li><math>\mathcal{{math|1=𝒰U}(∅\varnothing, ''N'') = ''F''}}.</math></li>
<li>{{<math|𝒰>\mathcal{U}(''G'', ''N'') - 𝒰\mathcal{U}(''G'', ''N'') ⊆\subseteq 𝒰\mathcal{U}(''G'', ''N'' - ''N'')}}.</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}}</li>
<li>{{<math|𝒰>\mathcal{U}(''G'', ''M'') + 𝒰\mathcal{U}(''G'', ''N'') ⊆\subseteq 𝒰\mathcal{U}(''G'', ''M'' + ''N'')}}.</math>{{sfn|Jarchow|1981|pp=43-55}}</li>
<li>For any family <math>\mathcal{S}</math> of subsets of <math>T,</math> and any family <math>\mathcal{{M}</math> of neighborhoods of the origin in <math>Y,</math>{{sfn|1Narici|Beckenstein|2011|pp=𝒰19-45}} <math display="block">\mathcal{U}\left(\bigcup_{{underset|S ∈\in 𝒮|\mathcal{{big|∪}}S}} ''S'', ''N''\right) = \bigcap_{{underset|S ∈\in 𝒮|\mathcal{{big|∩}}S}} 𝒰\mathcal{U}(''S'', ''N'') \qquad \text{ and } \qquad \mathcal{U}.\left(G, \bigcap_{M \in \mathcal{sfn|Narici|Beckenstein|2011|ppM}} M\right) =19-45 \bigcap_{M \in \mathcal{M}} \mathcal{U}(G, M).</math></li>
<li>For any family <math>\mathcal{M}</math> of neighborhoods of 0 in <math>Y,</math> {{math|1=𝒰(''G'', {{underset|M ∈ ℳ|{{big|∩}}}} ''M'') = {{underset|M ∈ ℳ|{{big|∩}}}} 𝒰(''G'', ''M'')}}.{{sfn|Narici|Beckenstein|2011|pp=19-45}}</li>
</ul>
 
=== <math>\mathcal{G}</math>-topology ===
 
Then the family
Then the set {{math|{𝒰(''G'', ''N'') : ''G'' ∈ 𝒢, ''N'' ∈ 𝒩}}} forms a [[Neighbourhood system|neighborhood basis]]<ref>Note that each set <math>\mathcal{U}(G, N)</math> is a neighborhood of the origin for this topology, but it is not necessarily an ''open'' neighborhood of the origin.</ref> ▼
<math display="block">\{ \mathcal{U}(G, N) : G \in \mathcal{G}, N \in \mathcal{N} \}</math>
at the origin for a unique translation-invariant topology on <math>F,</math> where this topology is ''not'' necessarily a vector topology (i.e. it might not make <math>F</math> into a TVS). ▼
▲Then the set {{math|{𝒰(''G'', ''N'') : ''G'' ∈ 𝒢, ''N'' ∈ 𝒩}}} forms a [[Neighbourhood system|neighborhood basis]]<ref>Note that each set <math>\mathcal{U}(G, N)</math> is a neighborhood of the origin for this topology, but it is not necessarily an ''open'' neighborhood of the origin.</ref>
This topology does not depend on the neighborhood basis {{math|𝒩}} that was chosen and it is known as the '''topology of uniform convergence on the sets in <math>\mathcal{G}</math>''' or as the '''<math>\mathcal{G}</math>-topology'''.{{sfn|Schaefer|Wolff|1999|pp=79-88}} ▼
▲at the origin for a unique translation-invariant topology on <math>F,</math> where this topology is ''{{em|not''}} necessarily a vector topology (i.e. it might not make <math>F</math> into a TVS).
▲This topology does not depend on the neighborhood basis <math>\mathcal{{N}</math|𝒩}}> that was chosen and it is known as the '''topology of uniform convergence on the sets in <math>\mathcal{G}</math>''' or as the '''<math>\mathcal{G}</math>-topology'''.{{sfn|Schaefer|Wolff|1999|pp=79-88}}
However, this name is frequently changed according to the types of sets that make up <math>\mathcal{G}</math> (e.g. the "topology of uniform convergence on compact sets" or the "topology of compact convergence", see the footnote for more details<ref>In practice, <math>\mathcal{G}</math> usually consists of a collection of sets with certain properties and this name is changed appropriately to reflect this set so that if, for instance, <math>\mathcal{G}</math> is the collection of compact subsets of <math>T</math> (and <math>T</math> is a topological space), then this topology is called the topology of uniform convergence on the compact subsets of <math>T.</math></ref>).
 
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{{Math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|pp=79-88}}{{sfn|Jarchow|1981|pp=43-55}}|math_statement=
The <math>\mathcal{G}</math>-topology on <math>F</math> is compatible with the vector space structure of <math>F</math> if and only if every <math>G \in \mathcal{G}</math> is <math>F</math>-bounded;
that is, if and only if for every <math>G \in \mathcal{G}</math> and every <math>f \in F,</math> {{<math|''>f'' (''G'')}}</math> is [[Bounded set (topological vector space)|bounded]] in <math>Y.</math>
}}
 
==== Nets and uniform convergence ====
 
:'''Definition''':{{sfn|Jarchow|1981|pp=43-55}} Let <math>f \in F</math> and let {{<math|1=''f''<sub>•</sub>f_{\bull} = \left(''f''<sub>''i''</sub>f_i\right)<sub>''_{i'' ∈\in ''I''}</submath>}} be a [[Net (mathematics)|net]] in <math>F.</math> Then for any subset <math>G</math> of <math>T,</math> say that {{<math|''f''<sub>•f_{\bull}</submath>}} '''converges uniformly to {{mvar|<math>f}}</math> on <math>G</math>''' if for every <math>N \in \mathcal{N}</math> there exists some {{<math|''i''<sub>0i_0 \in I</submath> ∈ ''I''}} such that for every {{<math|''>i'' ∈\in ''I''}}</math> satisfying {{<math|''>i'' ≥\geq ''i''<sub>0i_0,I</submath>}}, {{<math|''f''<sub>''i''</sub>f_i - ''f'' ∈\in 𝒰\mathcal{U}(''G'', ''N'')}}</math> (or equivalently, {{<math|''f''<sub>''i''</sub>f_i(''g'') - ''f'' (''g'') ∈\in ''N''}}</math> for every {{<math|''>g'' ∈\in ''G''}}</math>).
 
{{Math theorem|name=Theorem{{sfn|Jarchow|1981|pp=43-55}}|math_statement=
If <math>f \in F</math> and if {{<math|1=''f''<sub>•</sub>f_{\bull} = \left(''f''<sub>''i''</sub>f_i\right)<sub>''_{i'' ∈\in ''I''}</submath>}} is a net in <math>F,</math> then <math>f_{{math|''\bull} \to f''<sub>•</submath> → ''f''}} in the <math>\mathcal{G}</math>-topology on <math>F</math> if and only if for every <math>G \in \mathcal{G},</math> {{math|''f''<submath>•f_{\bull}</submath>}} converges uniformly to {{mvar|<math>f}}</math> on <math>G.</math>
}}
 
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;Local convexity
 
If <math>Y</math> is [[locally convex]] then so is the <math>\mathcal{G}</math>-topology on <math>F</math> and if {{<math|>\left(''p''<sub>''i''</sub>p_i\right)<sub>''_{i'' ∈\in ''I''}</submath>}} is a family of continuous seminorms generating this topology on <math>Y</math> then the <math>\mathcal{G}</math>-topology is induced by the following family of seminorms:
<math display="block">p_{G,i}(f) := \sup_{x \in G} p_i(f(x)),</math>
:{{math|''p''<sub>''G'',''i''</sub>(''f'') {{=}}}} {{underset|{{math|''x'' ∈ ''G''}}|sup}} {{math|''p''<sub>''i''</sub>(''f''(''x''))}},
as <math>G</math> varies over <math>\mathcal{G}</math> and {{mvar|<math>i}}</math> varies over {{mvar|<math>I}}</math>.{{sfn|Schaefer|Wolff|1999|p=81}}
 
;Hausdorffness
 
If <math>Y</math> is [[Hausdorff space|Hausdorff]] and {{<math|1=''>T'' = \bigcup_{{underset|''G'' ∈\in 𝒢|\mathcal{{big|∪}}G}} ''G''}}</math> then the <math>\mathcal{G}</math>-topology on <math>F</math> is Hausdorff.{{sfn|Jarchow|1981|pp=43-55}}
 
Suppose that <math>T</math> is a topological space.
If <math>Y</math> is [[Hausdorff space|Hausdorff]] and <math>F</math> is the vector subspace of <math>Y^T</math> consisting of all continuous maps that are bounded on every <math>G \in \mathcal{G}</math> and if {{<math|>\bigcup_{{underset|''G'' ∈\in 𝒢|\mathcal{{big|∪}}G}} ''G''}}</math> is dense in <math>T</math> then the <math>\mathcal{G}</math>-topology on <math>F</math> is Hausdorff.
 
;Boundedness
 
A subset <math>H</math> of <math>F</math> is [[Bounded set (topological vector space)|bounded]] in the <math>\mathcal{G}</math>-topology if and only if for every <math>G \in \mathcal{G},</math> {{<math|1=''>H''(''G'') := \bigcup_{{underset|''h'' ∈\in ''H''|{{big|∪}}}} ''h''(''G'')}}</math> is bounded in <math>Y.</math>{{sfn|Schaefer|Wolff|1999|p=81}}
 
=== Examples of 𝒢-topologies ===
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The topology of pointwise convergence on <math>F</math> is identical to the subspace topology that <math>F</math> inherits from <math>Y^T</math> when <math>Y^T</math> is endowed with the usual [[product topology]].
 
If <math>X</math> is a non-trivial [[Completely regular space|completely regular]] Hausdorff topological space and {{<math|>C(''X'')}}</math> is the space of all real (or complex) valued continuous functions on <math>X,</math> the topology of pointwise convergence on {{<math|>C(''X'')}}</math> is [[Metrizable TVS|metrizable]] if and only if <math>X</math> is countable.{{sfn|Jarchow|1981|pp=43-55}}
 
== 𝒢-topologies on spaces of continuous linear maps ==
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<math>\mathcal{G}</math> will be a non-empty collection of subsets of <math>X</math> [[Directed set|directed]] by inclusion.
 
:'''Notation''': <math>L(X; Y)</math> will denote the vector space of all continuous linear maps from <math>X</math> into <math>Y.</math> If <math>L(X; Y)</math> is given the <math>\mathcal{G}</math>-topology inherited from {{<math|''>Y''<sup>''^X''</supmath>}} then this space with this topology is denoted by <math>L_{\mathcal{math|L<sub>𝒢</sub>G}}(''X'',; ''Y'')}}</math>.
 
:'''Notation''': The [[Dual space#Continuous dual space|continuous dual space]] of a topological vector space <math>X</math> over the field <math>\mathbb{F}</math> (which we will assume to be [[real numbers|real]] or [[complex numbers]]) is the vector space <math>L(X; \mathbb{F})</math> and is denoted by {{<math|''>X''^{{big|{{'}}}}}\prime}</math>.
 
The <math>\mathcal{G}</math>-topology on <math>L(X; Y)</math> is compatible with the vector space structure of <math>L(X; Y)</math> if and only if for all <math>G \in \mathcal{G}</math> and all {{<math|''>f'' ∈\in L(''X''; ''Y'')}}</math> the set {{<math|''>f''(''G'')}}</math> is bounded in <math>Y,</math> which we will assume to be the case for the rest of the article.
Note in particular that this is the case if <math>\mathcal{G}</math> consists of [[Bounded set (topological vector space)|(von-Neumann) bounded]] subsets of <math>X.</math>
 
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;Assumptions that guarantee a vector topology
 
:'''Assumption''' (<math>\mathcal{G}</math> is directed): <math>\mathcal{G}</math> will be a non-empty collection of subsets of <math>X</math> [[Directed set|directed]] by (subset) inclusion. That is, for any <math>G, H \in \mathcal{G},</math> there exists <math>K \in \mathcal{G}</math> such that {{<math|''>G'' ∪\cup ''H'' ⊆\subseteq ''K''}}</math>.
 
The above assumption guarantees that the collection of sets <math>\mathcal{U}(G, N)</math> forms a [[filter base]].
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Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdonsome.
 
:'''Assumption''' (<math>N \in \mathcal{N}</math> are balanced): {{<math|𝒩}>\mathcal{N}</math> is a neighborhoods basis of 0the origin in <math>Y</math> that consists entirely of [[Balanced set|balanced]] sets.
 
The following assumption is very commonly made because it will guarantee that each set <math>\mathcal{U}(G, N)</math> is absorbing in <math>L(X; Y).</math>
Line 153 ⟶ 152:
 
Some authors (e.g. Trèves) require that <math>\mathcal{G}</math> be directed under subset inclusion and that it satisfy the following condition:
:If <math>G \in \mathcal{G}</math> and {{mvar|<math>s}}</math> is a scalar then there exists a <math>H \in \mathcal{G}</math> such that {{<math|''sG''>s ⊆G \subseteq ''H''}}.</math>
If <math>\mathcal{G}</math> is a [[bornology]] on <math>X,</math> which is often the case, then these axioms are satisfied.
If <math>\mathcal{G}</math> is a [[saturated family]] of [[Bounded set (topological vector space)|bounded]] subsets of <math>X</math> then these axioms are also satisfied.
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;Hausdorffness
 
:'''Definition''':{{sfn|Schaefer|Wolff|1999|p=80}} If <math>T</math> is a TVS then we say that <math>\mathcal{G}</math> is '''total in <math>T</math>''' if the [[linear span]] of {{<math|>\bigcup_{{underset|''G'' ∈\in 𝒢|\mathcal{{big|∪}}G}} ''G''}}</math> is dense in <math>T.</math>
 
If <math>F</math> is the vector subspace of <math>Y^T</math> consisting of all continuous linear maps that are bounded on every <math>G \in \mathcal{G},</math> then the <math>\mathcal{G}</math>-topology on <math>F</math> is Hausdorff if <math>Y</math> is Hausdorff and <math>\mathcal{G}</math> is total in <math>T.</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}
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;Completeness
 
For the following theorems, suppose that <math>X</math> is a topological vector space and <math>Y</math> is a [[locally convex]] Hausdorff spaces and <math>\mathcal{G}</math> is a collection of bounded subsets of <math>X</math> that covers <math>X,</math> is directed by subset inclusion, and satisfies the following condition: if <math>G \in \mathcal{G}</math> and {{mvar|<math>s}}</math> is a scalar then there exists a <math>H \in \mathcal{G}</math> such that {{<math|''sG''>s ⊆G \subseteq ''H''}}.</math>
 
<ul>
<li>{{math|L<sub>𝒢</submath>L_{\mathcal{G}}(''X''; ''Y'')}}</math> is complete if
{{ordered list|
|<math>X</math> is locally convex and Hausdorff,
|<math>Y</math> is complete, and
|whenever {{<math|''>u'' : ''X'' →\to ''Y''}}</math> is a linear map then {{mvar|<math>u}}</math> restricted to every set <math>G \in \mathcal{G}</math> is continuous implies that {{mvar|<math>u}}</math> is continuous,
}}</li>
<li>If <math>X</math> is a Mackey space then {{math|L<sub>𝒢</submath>L_{\mathcal{G}}(''X''; ''Y'')}}</math>is complete if and only if both <math>X^{\prime}_{\mathcal{G}}</math> and <math>Y</math> are complete.</li>
<li>If <math>X</math> is [[Barrelled space|barrelled]] then {{math|L<sub>𝒢</submath>L_{\mathcal{G}}(''X''; ''Y'')}}</math> is Hausdorff and [[quasi-complete]].</li>
<li>Let <math>X</math> and <math>Y</math> be TVSs with <math>Y</math> [[quasi-complete]] and assume that (1) <math>X</math> is [[barreled space|barreled]], or else (2) <math>X</math> is a [[Baire space]] and <math>X</math> and <math>Y</math> are locally convex. If <math>\mathcal{G}</math> covers <math>X</math> then every closed equicontinuous subset of <math>L(X; Y)</math> is complete in {{<math|L<sub>𝒢</sub>L_{\mathcal{G}}(''X''; ''Y'')}}</math> and {{math|L<sub>𝒢</submath>L_{\mathcal{G}}(''X''; ''Y'')}}</math> is quasi-complete.{{sfn|Schaefer|Wolff|1999|p=83}}</li>
<li>Let <math>X</math> be a [[bornological space]], <math>Y</math> a locally convex space, and <math>\mathcal{G}</math> a family of bounded subsets of <math>X</math> such that the range of every null sequence in <math>X</math> is contained in some <math>G \in \mathcal{G}.</math> If <math>Y</math> is [[quasi-complete]] (resp. complete) then so is {{math|L<sub>𝒢</submath>L_{\mathcal{G}}(''X''; ''Y'')}}</math>.{{sfn|Schaefer|Wolff|1999|p=117}}</li>
</ul>
 
Line 187 ⟶ 186:
Then the following are equivalent:{{sfn|Schaefer|Wolff|1999|p=81}}
<ol>
<li><math>H</math> is [[Bounded set (topological vector space)|bounded]] in {{math|L<sub>𝒢</submath>L_{\mathcal{G}}(''X''; ''Y'')}}</math>;</li>
<li>For every <math>G \in \mathcal{G},</math> {{<math|1=''>H''(''G'') := \bigcup_{{underset|''h'' ∈\in ''H''|{{big|∪}}}} ''h''(''G'')}}</math> is bounded in <math>Y</math>;{{sfn|Schaefer|Wolff|1999|p=81}}</li>
<li>For every neighborhood {{mvar|<math>V}}</math> of 0the origin in <math>Y</math> the set {{<math|>\bigcap_{{underset|''h'' ∈\in ''H''|{{big|∩}}}} ''h''<sup>−1^{-1}(V)</supmath>(''V'')}} [[Absorbing set|absorbs]] every <math>G \in \mathcal{G}.</math></li>
</ol>
 
Furthermore,
<ul>
<li>If <math>X</math> and <math>Y</math> are locally convex Hausdorff space and if <math>H</math> is bounded in {{<math|L<sub>𝜎</sub>L_{\sigma}(''X''; ''Y'')}}</math> (i.e. pointwise bounded or simply bounded) then it is bounded in the topology of uniform convergence on the convex, balanced, bounded, complete subsets of <math>X.</math>{{sfn|Schaefer|Wolff|1999|p=82}}</li>
<li>If <math>X</math> and <math>Y</math> are locally convex Hausdorff spaces and if <math>X</math> is quasi-complete (i.e. closed and bounded subsets are complete), then the bounded subsets of <math>L(X; Y)</math> are identical for all <math>\mathcal{G}</math>-topologies where <math>\mathcal{G}</math> is any family of bounded subsets of <math>X</math> covering <math>X.</math>{{sfn|Schaefer|Wolff|1999|p=82}}</li>
<li>If <math>\mathcal{G}</math> is any collection of bounded subsets of <math>X</math> whose union is total in <math>X</math> then every equicontinuous subset of <math>L(X; Y)</math> is bounded in the <math>\mathcal{G}</math>-topology.{{sfn|Schaefer|Wolff|1999|p=83}}</li>
Line 202 ⟶ 201:
{| class="wikitable"
|-
! <math>\mathcal{{math|𝒢G} ⊆\subseteq 𝒫\wp(''X'')}}</math> ("topology of uniform convergence on ...")
! Notation
! Name ("topology of...")
Line 208 ⟶ 207:
|-
| finite subsets of <math>X</math>
| {{<math|L<sub>σ</sub>L_{\sigma}(''X''; ''Y'')}}</math>
| pointwise/simple convergence
| topology of simple convergence
Line 218 ⟶ 217:
|-
| compact convex subsets of <math>X</math>
| {{<math|L<sub>γ</sub>L_{\gamma}(''X''; ''Y'')}}</math>
| compact convex convergence
|
|-
| compact subsets of <math>X</math>
| {{math|L<sub>c</submath>L_c(''X''; ''Y'')}}</math>
| compact convergence
|
|-
| bounded subsets of <math>X</math>
| {{math|L<sub>b</submath>L_b(''X''; ''Y'')}}</math>
| bounded convergence
| strong topology
|}
 
==== The topology of pointwise convergence {{<math|L<sub>σ</sub>L_{\sigma}(''X''; ''Y'')}}</math> ====
 
By letting <math>\mathcal{G}</math> be the set of all finite subsets of <math>X,</math> <math>L(X; Y)</math> will have the '''weak topology on <math>L(X; Y)</math>''' or '''the topology of pointwise convergence''' or '''the topology of simple convergence''' and <math>L(X; Y)</math> with this topology is denoted by {{<math|L<sub>𝜎</sub>L_{\sigma}(''X''; ''Y'')}}</math>.
Unfortunately, this topology is also sometimes called '''the strong operator topology''', which may lead to ambiguity;{{sfn|Narici|Beckenstein|2011|pp=371-423}} for this reason, this article will avoid referring to this topology by this name.
 
:'''Definition''': A subset of <math>L(X; Y)</math> is called '''simply bounded''' or '''weakly bounded''' if it is bounded in {{<math|L<sub>𝜎</sub>L_{\sigma}(''X''; ''Y'')}}</math>.
 
The weak-topology on <math>L(X; Y)</math> has the following properties:
<ul>
<li>If <math>X</math> is [[Separable space|separable]] (i.e. has a countable dense subset) and if <math>Y</math> is a metrizable topological vector space then every equicontinuous subset <math>H</math> of {{<math|L<sub>𝜎</sub>L_{\sigma}(''X''; ''Y'')}}</math> is metrizable; if in addition <math>Y</math> is separable then so is <math>H.</math>{{sfn|Schaefer|Wolff|1999|p=87}}
* So in particular, on every equicontinuous subset of <math>L(X; Y),</math> the topology of pointwise convergence is metrizable.</li>
<li>Let {{math|''Y''<supmath>''Y^X''</supmath>}} denote the space of all functions from <math>X</math> into <math>Y.</math> If <math>L(X; Y)</math> is given the topology of pointwise convergence then space of all linear maps (continuous or not) <math>X</math> into <math>Y</math> is closed in {{math|''Y''<supmath>''Y^X''</supmath>}}.
* In addition, <math>L(X; Y)</math> is dense in the space of all linear maps (continuous or not) <math>X</math> into <math>Y.</math></li>
<li>Suppose <math>X</math> and <math>Y</math> are locally convex. Any simply bounded subset of <math>L(X; Y)</math> is bounded when <math>L(X; Y)</math> has the topology of uniform convergence on convex, [[balanced set|balanced]], bounded, complete subsets of <math>X.</math> If in addition <math>X</math> is [[quasi-complete]] then the families of bounded subsets of <math>L(X; Y)</math> are identical for all <math>\mathcal{G}</math>-topologies on <math>L(X; Y)</math> such that <math>\mathcal{G}</math> is a family of bounded sets covering <math>X.</math>{{sfn|Schaefer|Wolff|1999|p=82}}</li>
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</ul>
 
==== Compact convergence {{math|L<sub>c</submath>L_c(''X''; ''Y'')}}</math> ====
 
By letting <math>\mathcal{G}</math> be the set of all compact subsets of <math>X,</math> <math>L(X; Y)</math> will have '''the topology of compact convergence''' or '''the topology of uniform convergence on compact sets''' and <math>L(X; Y)</math> with this topology is denoted by {{math|L<sub>c</submath>L_c(''X''; ''Y'')}}</math>.
 
The topology of compact convergence on <math>L(X; Y)</math> has the following properties:
<ul>
<li>If <math>X</math> is a [[Fréchet space]] or a [[LF-space]] and if <math>Y</math> is a [[Complete metric space#Topologically complete spaces|complete]] locally convex Hausdorff space then {{math|L<sub>c</submath>L_c(''X''; ''Y'')}}</math> is complete.</li>
<li>On equicontinuous subsets of <math>L(X; Y),</math> the following topologies coincide:
* The topology of pointwise convergence on a dense subset of <math>X,</math>
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* The topology of compact convergence.
* The topology of precompact convergence.</li>
<li>If <math>X</math> is a [[Montel space]] and <math>Y</math> is a topological vector space, then {{<math|L<sub>c</sub>L_c(''X''; ''Y'')}}</math> and {{math|L<sub>b</submath>L_b(''X''; ''Y'')}}</math> have identical topologies.</li>
</ul>
 
==== Topology of bounded convergence {{math|L<sub>b</submath>L_b(''X''; ''Y'')}}</math> ====
 
By letting <math>\mathcal{G}</math> be the set of all bounded subsets of <math>X,</math> <math>L(X; Y)</math> will have '''the topology of bounded convergence on <math>X</math>''' or '''the topology of uniform convergence on bounded sets''' and <math>L(X; Y)</math> with this topology is denoted by {{math|L<sub>b</submath>L_b(''X''; ''Y'')}}</math>.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
 
The topology of bounded convergence on <math>L(X; Y)</math> has the following properties:
<ul>
<li>If <math>X</math> is a [[bornological space]] and if <math>Y</math> is a [[Complete metric space#Topologically complete spaces|complete]] locally convex Hausdorff space then {{math|L<sub>b</submath>L_b(''X''; ''Y'')}}</math> is complete.</li>
<li>If <math>X</math> and <math>Y</math> are both normed spaces then the topology on <math>L(X; Y)</math> induced by the usual operator norm is identical to the topology on {{math|L<sub>b</submath>L_b(''X''; ''Y'')}}</math>.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
* In particular, if <math>X</math> is a normed space then the usual norm topology on the continuous dual space {{<math|''>X''{{big|{^{'}}}}}\prime}</math> is identical to the topology of bounded convergence on {{<math|''>X''^{{big|{{'}}}}}\prime}</math>.</li>
<li>Every equicontinuous subset of <math>L(X; Y)</math> is bounded in {{math|L<sub>b</submath>L_b(''X''; ''Y'')}}</math>.</li>
</ul>
 
Line 291 ⟶ 290:
=== <math>\mathcal{G}</math>-topologies versus polar topologies ===
 
If <math>X</math> is a TVS whose [[Bounded set (topological vector space)|bounded]] subsets are exactly the same as its ''{{em|weakly''}} bounded subsets (e.g. if <math>X</math> is a Hausdorff locally convex space), then a <math>\mathcal{G}</math>-topology on {{<math|''>X''{^{big|{{'}}}}}\prime}</math> (as defined in this article) is a [[polar topology]] and conversely, every polar topology if a <math>\mathcal{G}</math>-topology.
Consequently, in this case the results mentioned in this article can be applied to polar topologies.
 
However, if <math>X</math> is a TVS whose bounded subsets are ''{{em|not''}} exactly the same as its ''{{em|weakly''}} bounded subsets, then the notion of "bounded in <math>X</math>" is stronger than the notion of "{{<math|σ>\sigma\left(''X'', ''X''^{{big|{{'}}}\prime}\right)}}</math>-bounded in <math>X</math>" (i.e. bounded in <math>X</math> implies {{<math|σ>\sigma\left(''X'', ''X''^{{big|{{'}}}\prime}\right)}}</math>-bounded in <math>X</math>) so that a <math>\mathcal{G}</math>-topology on {{<math|''>X''{^{big|{{'}}}}}\prime}</math> (as defined in this article) is ''{{em|not''}} necessarily a polar topology.
One important difference is that polar topologies are always locally convex while <math>\mathcal{G}</math>-topologies need not be.
 
Line 304 ⟶ 303:
Suppose that <math>X</math> is a TVS whose bounded subsets are the same as its weakly bounded subsets.
 
:'''Notation''': If {{<math|𝛥>\Delta(''Y'', ''X'')}}</math> denotes a polar topology on <math>Y</math> then <math>Y</math> endowed with this topology will be denoted by {{<math|''Y''<sub>𝛥Y_{\Delta(''Y'', ''X'')}</submath>}} or simply {{<math|''Y''<sub>𝛥Y_{\Delta}</submath>}} (e.g. for {{<math|σ>\sigma(''Y'', ''X'')}}</math> we'd would have {{<math|𝛥>\Delta {{=}} σ}}\sigma</math> so that {{<math|''Y''<sub>σY_{\sigma(''Y'', ''X'')}</submath>}} and {{<math|''Y''<sub>σY_{\sigma}</submath>}} all denote <math>Y</math> with endowed with {{<math|σ>\sigma(''Y'', ''X'')}}</math>).
 
{| class="wikitable"
|-
! ><math>\mathcal{{math|𝒢G} ⊆\subseteq 𝒫\wp(''X'')}}</math><br/>("topology of uniform convergence on ...")
! Notation
! Name ("topology of...")
Line 314 ⟶ 313:
|-
| finite subsets of <math>X</math>
| {{<math|σ>\sigma(''Y'', ''X'')}}</math><br/>{{<math|>s(''Y'', ''X'')}}</math>
| pointwise/simple convergence
| [[Weak topology|weak/weak* topology]]
|-
| {{<math|σ>\sigma(''X'', ''Y'')}}</math>-compact [[Absolutely convex set|disk]]s
| {{<math|τ>\tau(''Y'', ''X'')}}</math>
|
| [[Mackey topology]]
|-
| {{<math|σ>\sigma(''X'', ''Y'')}}</math>-compact convex subsets
| {{<math|γ>\gamma(''Y'', ''X'')}}</math>
| compact convex convergence
|
|-
| {{<math|σ>\sigma(''X'', ''Y'')}}</math>-compact subsets<br/>(or balanced {{<math|σ>\sigma(''X'', ''Y'')}}</math>-compact subsets)
| {{<math|>c(''Y'', ''X'')}}</math>
| compact convergence
|
|-
| {{<math|σ>\sigma(''X'', ''Y'')}}</math>-bounded subsets
| {{<math|>b(''Y'', ''X'')}}</math><br/>{{<math|𝛽>\beta(''Y'', ''X'')}}</math>
| bounded convergence
| [[Strong dual space|strong topology]]
Line 341 ⟶ 340:
== 𝒢-ℋ-topologies on spaces of bilinear maps ==
 
We will let {{<math|ℬ>\mathcal{B}(''X'', ''Y''; ''Z'')}}</math> denote the space of separately continuous bilinear maps and {{<math|>B(''X'', ''Y''; ''Z'')}} </math>denote the space of continuous bilinear maps, where <math>X, Y,</math> and <math>Z</math> are topological vector space over the same field (either the real or complex numbers).
In an analogous way to how we placed a topology on <math>L(X; Y)</math> we can place a topology on <math>\mathcal{{math|ℬB}(''X'', ''Y''; ''Z'')}}</math> and {{<math|>B(''X'', ''Y''; ''Z'')}}</math>.
 
Let <math>\mathcal{G}</math> (resp. <math>\mathcal{H}</math>) be a family of subsets of <math>X</math> (resp. <math>Y</math>) containing at least one non-empty set.
Let <math>\mathcal{G} \times \mathcal{H}</math> denote the collection of all sets <math>G \times H</math> where <math>G \in \mathcal{G},</math> <math>H \in \mathcal{H}.</math>
We can place on {{math|''Z''<supmath>''Z^{X'' ×\times ''Y''}</supmath>}} the <math>\mathcal{G} \times \mathcal{H}</math>-topology, and consequently on any of its subsets, in particular on {{<math|>B(''X'', ''Y''; ''Z'')}} </math>and on {{<math|ℬ>\mathcal{B}(''X'', ''Y''; ''Z'')}}</math>.
This topology is known as the '''<math>\mathcal{G}-\mathcal{H}</math>-topology''' or as the '''topology of uniform convergence on the products <math>G \times H</math> of <math>\mathcal{G} \times \mathcal{H}</math>'''.
 
However, as before, this topology is not necessarily compatible with the vector space structure of {{<math|ℬ>\mathcal{B}(''X'', ''Y''; ''Z'')}}</math> or of {{<math|>B(''X'', ''Y''; ''Z'')}} </math>without the additional requirement that for all bilinear maps, <math>b</math> in this space (that is, in {{<math|ℬ>\mathcal{B}(''X'', ''Y''; ''Z'')}}</math> or in {{<math|>B(''X'', ''Y''; ''Z'')}}</math>) and for all <math>G \in \mathcal{G}</math> and <math>H \in \mathcal{H},</math> the set {{<math|>b(''G'', ''H'')}}</math> is bounded in <math>X.</math>
If both <math>\mathcal{G}</math> and <math>\mathcal{H}</math> consist of bounded sets then this requirement is automatically satisfied if we are topologizing {{<math|>B(''X'', ''Y''; ''Z'')}} </math>but this may not be the case if we are trying to topologize <math>\mathcal{{math|ℬB}(''X'', ''Y''; ''Z'')}}</math>.
The <math>\mathcal{G}-\mathcal{H}</math>-topology on {{<math|ℬ>\mathcal{B}(''X'', ''Y''; ''Z'')}}</math> will be compatible with the vector space structure of <math>\mathcal{{math|ℬB}(''X'', ''Y''; ''Z'')}}</math> if both <math>\mathcal{G}</math> and <math>\mathcal{H}</math> consists of bounded sets and any of the following conditions hold:
* <math>X</math> and <math>Y</math> are barrelled spaces and <math>Z</math> is locally convex.
* <math>X</math> is a [[F-space]], <math>Y</math> is metrizable, and <math>Z</math> is Hausdorff, in which case <math>\mathcal{{math|1=ℬB}(''X'', ''Y''; ''Z'') = B(''X'', ''Y''; ''Z'')}}.</math>
* <math>X, Y,</math> and <math>Z</math> are the strong duals of reflexive Fréchet spaces.
* <math>X</math> is normed and <math>Y</math> and <math>Z</math> the strong duals of reflexive Fréchet spaces.
Line 360 ⟶ 359:
{{Main|Injective tensor product}}
 
Suppose that <math>X, Y,</math> and <math>Z</math> are locally convex spaces and let {{<math|𝒢>\mathcal{G}^{'}}}\prime}</math> and {{<math|ℋ>\mathcal{H}^{'}}}\prime}</math> be the collections of equicontinuous subsets of {{<math|''>X''^{{big|{{'}}}}}\prime}</math> and {{<math|''Y''>X^{{big|{{'}}}}}\prime}</math>, respectively.
Then the {{<math|𝒢>\mathcal{G}^{'}\prime}-ℋ\mathcal{H}^{'}}}\prime}</math>-topology on <math>\mathcal{B}\left(X^{\prime}_{b\left(X^{\prime}, X\right)}, Y^{\prime}_{b\left(X^{\prime}, X\right)}; Z\right)</math> will be a topological vector space topology.
This topology is called the ε-topology and <math>\mathcal{B}\left(X^{\prime}_{b\left(X^{\prime}, X\right)}, Y_{b\left(X^{\prime}, X\right)}; Z\right)</math> with this topology it is denoted by <math>\mathcal{B}_{\epsilon}\left(X^{\prime}_{b\left(X^{\prime}, X\right)}, Y^{\prime}_{b\left(X^{\prime}, X\right)}; Z\right)</math> or simply by <math>\mathcal{B}_{\epsilon}\left(X^{\prime}_{b}, Y^{\prime}_{b}; Z\right).</math>
 
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These spaces have the following properties:
* If <math>X</math> and <math>Y</math> are locally convex Hausdorff spaces then <math>\mathcal{B}_{\varepsilon}\left(X^{\prime}_{\sigma}, Y^{\prime}_{\sigma}\right)</math> is complete if and only if both <math>X</math> and <math>Y</math> are complete.
* If <math>X</math> and <math>Y</math> are both normed (orrespectively, both Banach) then so is <math>\mathcal{B}_{\epsilon}\left(X^{\prime}_{\sigma}, Y^{\prime}_{\sigma}\right)</math>
 
== See also ==
Line 392 ⟶ 391:
 
== References ==
{{Reflist}}
 
{{reflist|group=note}}
==Bibliography==▼
{{reflist|group=proof}}
{{reflist}}
 
▲== Bibliography ==
 
* {{Jarchow Locally Convex Spaces}}
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn|Khaleelulla|{{{year| 1982 }}}|p=}} -->
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